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Determine the order N of a Butterworth filter for which Amax = 1 dB,

Determine the order N of a Butterworth filter for which Amax = 1 dB,

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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EXERCISES D17.6 Determine the order N of a Butterworth filter for which A =1 dB, w lo, =1.5, and A = 30 dB. What is the actual value of minimum stopband attenuation realized? If Amin is to be exactly 30 dB, to what value can Ama be reduced? Ans. N = 11; Amin = 32.87 dB; 0.54 dB 17.7 Find the natural modes and the transfer function of a Butterworth filter with w,=1 rad/s, Amax =3 dB (e- 1), and N = 3. Ans. –0.5 +j /3/2 and –1; T(s) = 1/(s+1)(s +s+1) 17.8 Observe that Eq. (17.18) can be used to find the frequencies in the passband at which |T| is at its peaks and at its valleys. (The peaks are reached when the cos [ ] term is zero, and the valleys correspond to the cos [ ] term equal to unity.) Find these frequencies for a fifth-order filter. Ans. Peaks at w = 0, 0.59w,, and 0.95a,; the valleys at w = 0.31@, and 0.81@, D17.9 Find the attenuation provided at w = 20, by a seventh-order Chebyshev filter with a 0.5-dB passband ripple. If the passband ripple is allowed to increase to 1 dB, by how much does the stopband attenuation increase? Ans. 64.9 dB; 3.3 dB D17.10 It is required to design a low-pass filter having f, = 1 kHz, A = 1 dB,f, = 1.5 kHz, A = 50 dB. (a) Find the required order of a Chebyshev filter. What is the excess stopband attenuation obtained? (b) Repeat for a Butterworth filter. Ans. (a) N = 8, 5 dB; (b) N = 16, 0.5 dB
Transcribed Image Text:EXERCISES D17.6 Determine the order N of a Butterworth filter for which A =1 dB, w lo, =1.5, and A = 30 dB. What is the actual value of minimum stopband attenuation realized? If Amin is to be exactly 30 dB, to what value can Ama be reduced? Ans. N = 11; Amin = 32.87 dB; 0.54 dB 17.7 Find the natural modes and the transfer function of a Butterworth filter with w,=1 rad/s, Amax =3 dB (e- 1), and N = 3. Ans. –0.5 +j /3/2 and –1; T(s) = 1/(s+1)(s +s+1) 17.8 Observe that Eq. (17.18) can be used to find the frequencies in the passband at which |T| is at its peaks and at its valleys. (The peaks are reached when the cos [ ] term is zero, and the valleys correspond to the cos [ ] term equal to unity.) Find these frequencies for a fifth-order filter. Ans. Peaks at w = 0, 0.59w,, and 0.95a,; the valleys at w = 0.31@, and 0.81@, D17.9 Find the attenuation provided at w = 20, by a seventh-order Chebyshev filter with a 0.5-dB passband ripple. If the passband ripple is allowed to increase to 1 dB, by how much does the stopband attenuation increase? Ans. 64.9 dB; 3.3 dB D17.10 It is required to design a low-pass filter having f, = 1 kHz, A = 1 dB,f, = 1.5 kHz, A = 50 dB. (a) Find the required order of a Chebyshev filter. What is the excess stopband attenuation obtained? (b) Repeat for a Butterworth filter. Ans. (a) N = 8, 5 dB; (b) N = 16, 0.5 dB
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